Global economic growth and agricultural land conversion under uncertain productivity improvements in agriculture

Transcription

1 University of Neuchatel Institute of Economic Research IRENE Working paper Global economic growth and agricultural land conversion under uncertain productivity improvements in agriculture Bruno Lanz Simon Dietz Tim Swanson

2 Global economic growth and agricultural land conversion under uncertain productivity improvements in agriculture Bruno Lanz Simon Dietz Tim Swanson This version: May 2017 Abstract We study how stochasticity in the evolution of agricultural productivity interacts with economic and population growth at the global level. We use a two-sector Schumpeterian model of growth, in which a manufacturing sector produces the traditional consumption good and an agricultural sector produces food to sustain contemporaneous population. Agriculture demands land as an input, itself treated as a scarce form of capital. In our model both population and sectoral technological progress are endogenously determined, and key technological parameters of the model are structurally estimated using data on world GDP, population, cropland and technological progress. Introducing random shocks to the evolution of total factor productivity in agriculture, we show that uncertainty optimally requires more land to be converted into agricultural use as a hedge against production shortages, and that it significantly affects both optimal consumption and population trajectories. Keywords: Agricultural productivity; Economic growth; Endogenous innovations; Environmental constraints; Food security; Global population; Land conversion; Stochastic control. JEL Classification numbers: O11, O13, O31, J11, C61, Q16, Q24. We thank David Laborde, Guy Meunier, Pietro Peretto, John Reilly, David Simpson, Simone Valente and Marty Weitzman, as well as seminar participants at ETH Zürich, INRA-ALISS and MIT for comments and discussions. Excellent research assistance was provided by Ozgun Haznedar. The research leading to these results has received funding from the European Union s Seventh Framework Programme FP7/ under Grant Agreement Number FOODSECURE. The authors alone are responsible for any omissions or deficiencies. Neither the FOODSECURE project and any of its partner organizations, nor any organization of the European Union or European Commission, are accountable for the content of the article. Corresponding author. University of Neuchâtel, Department of Economics and Business, Switzerland. Mail: A.-L. Breguet 2, CH-2000 Neuchâtel, Switzerland; Tel: ; bruno.lanz@unine.ch. London School of Economics and Political Science, Grantham Research Institute on Climate Change and the Environment, and Department of Geography and Environment, UK. Graduate Institute of International and Development Studies, Department of Economics and Centre for International Environmental Studies, Switzerland.

3 1 Introduction Between 1960 and 2010, the global population rose from about three to seven billion, more than it had increased in the previous two millennia (United Nations, 1999a), while real global GDP per capita increased by a factor of about 2.5 (World Bank, 2016). With more people to feed and a positive relationship between income per capita and food consumption per capita (Subramanian and Deaton, 1996; Tilman et al., 2011), aggregate food demand increased significantly. Over the same 50 years, however, agricultural production almost tripled, mostly on account of a sustained increase in agricultural productivity (Alexandratos and Bruinsma, 2012), with the result that food did not become more scarce, globally on aggregate (Alston and Pardey, 2014). Turning to the future, the global population is projected to continue expanding by several billion likely reaching 10 billion before 2060 (United Nations, 2015) and global GDP per capita might double by mid-century (Clarke et al., 2014). Hence further improvements in agricultural productivity will need to take place, driven by innovation and technology adoption. In this paper we study how uncertainty and variability in agricultural output affect the ability to feed a large, growing and increasingly rich global population. As we show in Figure 1, global average total factor productivity (TFP) growth in agriculture has been around one per cent per year over the period 1960 to 2010, contributing greatly to meeting the increase in food demand. But it also shows that there has been large variation in growth rates across regions and over time, ranging from -17 to +20 per cent per year. 1 Weather variability is one cause of the stochasticity in the historical agricultural TFP series. As Auffhammer and Schlenker (2014) observe in their review, the relationship between yields and weather, specifically temperature, is highly nonlinear and concave (also see Schlenker and Roberts, 2009). Consequently extreme heat over the growing season is a strong predictor of crop yields. Anthropogenic climate change is expected to change patterns of weather variability 1 Data on TFP growth are derived from Fuglie and Rada (2015) and FAO (2015). We use the growth accounting methodology of Fuglie and Rada (2015), which takes into account a broad set of inputs and aggregates TFP growth rates at the level of 27 macro regions. Compared to Fuglie and Rada (2015), who apply a Hodrick- Prescott filter to smooth year-on-year output fluctuations before calculating TFP, TFP growth rates reported in Figure 1 are based on raw (unsmoothed) output data from FAO (2015), with the purpose of highlighting variability of agricultural productivity growth. 1

4 Yearly TFP growth rate Figure 1: Total factor productivity growth in agriculture, Maximum change in TFP Average change in TFP Minimum change in TFP Notes: Plotted data on yearly TFP growth are derived from Fuglie and Rada (2015) and FAO (2015). Average change in TFP measures yearly growth rate of TFP averaged (without weights) across 27 macro regions defined in Fuglie and Rada (2015). Minimum and maximum yearly growth rates across regions are also reported. See footnote 1 for more details on the reported data. worldwide. The Intergovernmental Panel on Climate Change thinks that anthropogenic climate change is somewhere between very likely and virtually certain to result in more frequent incidences of extreme heat, depending on the definition and timescale, as well as increasing the frequency of other types of extreme weather, with varying, but generally lower, degrees of confidence (IPCC, 2013). In addition, structural models that do not incorporate weather variability nonetheless show that anthropogenic climate change is likely to reduce food supply and increase prices by way of gradual changes in average conditions (Nelson et al., 2014a). Other emerging sources of variability in agricultural TFP have also been put forward, including the loss of genetic and species diversity in farming systems (Di Falco, 2012), and increasing homogeneity of global food supplies (Khoury et al., 2014), making them potentially more vulnerable to covariate shocks. Inspired by these risks, some long-standing and some only now emerging, in this paper we study the socially optimal global response to the risk of negative shocks to global agricultural productivity. To do so we employ a stochastic version of a quantitative, two-sector endogenous growth model of the global economy that was introduced in Lanz et al. (forthcoming). This provides an integrated framework to study the joint evolution of global population, sectoral 2

5 technological progress, per-capita income, the demand for food, and agricultural land expansion (from a finite reserve of unconverted land). Specifically, the model distinguishes agriculture from other sectors of the economy (which produce a bundle of consumption goods) and treats both population and sectoral TFP as endogenous stock variables. The level of population in the model derives from preferences over fertility by a representative household (Barro and Becker, 1989), with fertility costs capturing two components. First, additional labor units demand food, and the level of per-capita food demand is proportional to income. In the model, food is produced by the agricultural sector, so that the evolution of agricultural productivity may act as a constraint on the evolution of population. A second fertility cost is the time needed to rear and educate children. Our model builds on the work of Galor and Weil (2000) by incorporating an increasing relationship between the level of technology in the economy and the cost of population increments. Technological progress raises education requirements and the demand for human capital, capturing the well-documented complementarity between technology and skills (Goldin and Katz, 1998). Given the explicit representation of fertility decisions and the demand for food associated with population and income growth, the model is well-suited to study the role of technology as a driver of global economic development. In the model, sectoral technological progress is endogenously determined by the Schumpeterian R&D model of Aghion and Howitt (1992), in which TFP growth is a function of labor hired by R&D firms. Thus, on the one hand technological progress in agriculture reduces the cost of producing food, and is an important driver of agricultural yields. In turn, agricultural technology improvements can alleviate Malthusian concerns about the finite land input. On the other hand, economy-wide technological progress implies a quantity-quality trade-off in fertility choices (through increasing education costs), and thus a slowdown of population growth (as per Galor and Weil, 2000). Taken together, technological progress is central to the development path generated by the model. As discussed in detail in Lanz et al. (forthcoming), we use simulation methods to structurally estimate key parameters of the model, minimizing the distance between observed and simulated trajectories for world GDP, population, TFP growth and agricultural land area. The estimated model closely replicates targeted data over the estimation period, and is also able to 3

6 replicate untargeted moments, such as the share of agriculture in world GDP and the growth rate of agricultural yields. In this article, we introduce uncertainty about the evolution of agricultural TFP in the coming years. Our objective is not to carry out an assessment of some specific uncertain event. Instead, our contribution is to provide an internally consistent picture of how uncertainty in the evolution of agricultural technology affects the socially optimal allocation of resources in a framework with endogenous land conversion, population, and R&D-based TFP growth. Our TFP shocks are therefore illustrative in nature, although they are calibrated to be within the same order of magnitude as shocks observed in the past. In the baseline, agricultural TFP growth starts at around one per cent per year in 2010 and declines thereafter. This implies that agricultural yields increase linearly, which is consistent with extrapolating data on trend growth in yields from the past several decades, particularly for the main grain crops (e.g. Alston et al., 2009; Godfray et al., 2010). Given the structure of productivity shocks we consider, there is a 73 per cent probability that this baseline situation prevails in If, on the other hand, negative productivity shocks occur, and realized shocks are permanent in the sense that they affect agricultural productivity in all subsequent periods, by 2030 there is a 24 per cent probability that agricultural TFP is around 10 per cent lower relative to its baseline value, a 3 per cent probability that it is 15 per cent lower, and a 0.1 per cent probability that it is more than 20 per cent lower. In the model, the socially optimal response to uncertain agricultural productivity shocks occurs in a number of key dimensions. First, given a risk of lower agricultural productivity in the future, more labor can be allocated to R&D, so as to speed up technological progress. Second, when a negative shock occurs, more primary factors can be allocated to agricultural production, specifically labor, capital and land. Here, increasing agricultural land area involves a decision to deplete a finite reserve base, so there is an intertemporal trade-off involved. Third, changes in agricultural productivity affect population growth through food availability. In particular, depreciation of agricultural technology increases the relative cost of food, with a negative effect on fertility decisions, so that agricultural productivity shocks affect equilibrium trajectories in the long run. Finally, per-capita consumption also adjusts downwards, as more resources are allocated to the agricultural sector at the expense of manufacturing production. 4

7 Results from the model indicate that the risk of negative shocks to agricultural TFP induces a substantial reallocation of resources relative to the baseline. The planner allocates more resources to agricultural R&D, but we find that, once a negative shock has occurred, agricultural TFP does not catch up with its baseline path. Thus in our framework it is too expensive for the planner to simply compensate lost agricultural TFP with supplementary R&D expenditure. Rather the planner expands use of other primary inputs to agriculture. But, since there is an opportunity cost of labor and capital (which are also used to produce the manufactured good), the main response of the planner is to increase the area of agricultural land. In addition, as technology shocks make food more expensive to produce, a second major implication is that population declines relative to the baseline. We carry out several extensions to the main analysis just described. First, we quantify how substitutability between land and other primary inputs to agriculture affects the finding that agricultural land is expanded. Our initial assumption is derived from the empirical work of Wilde (2013), which suggests an elasticity of substitution between land and other inputs of 0.6. We show that lower substitutability implies a significantly larger expansion of agricultural land in response to productivity shocks. Second, we shed light on the the role of per-capita income in the demand for food, by running a model in which food demand is simply proportional to population. This is equivalent to assuming a subsistence constraint, as considered by Strulik and Weisdorf (2008) for example, with zero income elasticity of food demand. Results suggest that agricultural land expansion is very similar, but the welfare cost of the productivity shocks is significantly larger. Finally, while our main set of runs is concerned with the occurrence of uncertain negative shocks to an otherwise increasing trend for agricultural productivity, the literature also raises the possibility of gradually stagnating and decreasing agricultural productivity (e.g. Alston et al., 2009). We therefore use the model to study a scenario in which trend agricultural productivity growth gradually slows and eventually goes into reverse. The model again suggests an extension of cropland area in order to compensate productivity losses. The remainder of the article is organized as follows. In Section 2, we discuss how our work relates to a number of strands of the literature. Section 3 provides an overview of the model and estimation procedure, and then describes how we introduce stochasticity in agricultural 5

8 productivity. Section 4 reports our main simulation results. Section 5 discusses implications of these results and sensitivity analysis. We close with some concluding comments in Section 6. 2 Relation to the literature Our work is related to at least two distinctive strands of literature that consider interactions between economic growth, food production and population development. First, our article is related to the seminal work of Galor and Weil (2000) and Jones (2001), which is aimed at fundamental understanding of the joint evolution of economic growth and population over the long run, and to Hansen and Prescott (2002), Strulik and Weisdorf (2008), Vollrath (2011), Sharp et al. (2012) and Strulik and Weisdorf (2014), who also consider the role of agriculture and land in growth. Related work by Bretschger (2013) and Peretto and Valente (2015) studies natural resource scarcity in a general, growth-theoretic setting. While our approach shares these theoretical underpinnings, it is distinctive in that key parameters of our quantitative model are structurally estimated, so that our model closely replicates observed trajectories over the past fifty years. In turn this allows us to investigate quantitatively the implications of stylized uncertainty about future technological progress. Second, our work is related to the literature on structural modeling of global agriculture, land use and food trade, which is used to estimate the impact of future climate change. Many of these models are brought together in the Agricultural Model Intercomparison and Improvement Project (AgMIP) (see in particular Nelson and Shively, 2014, and other papers in the same volume), which suggests that climate change could reduce global crop yields significantly and result in an increase of global cropland area. The models used to derive these results feature high-resolution sectoral and regional representations of agriculture and land use, which allows investigations into specific crops, regional impacts and trade. On the other hand, the evolution of key drivers determining global impacts (such as population, the demand for food, and agricultural yields) is exogenous to the simulations. By contrast, the model we formulate endogenizes global aggregate population, per-capita income, and technology, which allows us to study how these variables jointly respond to uncertainty about future agricultural productivity 6

9 growth. Our work also differs in how uncertainty about agricultural productivity is implemented. In structural modeling of climate impacts, different scenarios are used to introduce gradual changes in long-run average conditions, changes that are precisely calibrated on the outputs of climate and crop models. Our scenarios focus instead on short-run (but persistent) productivity shocks, which are calibrated to an order of magnitude on variability in past agricultural TFP, but are more illustrative in spirit. A paper in this line of research that is particularly close in spirit to our work is Cai et al. (2014), as they use a dynamic-stochastic partial equilibrium model of global land use to study the risk of an irreversible reduction in agricultural productivity. They show that, by 2100, this risk increases the demand for cropland globally, at the expense of valuable biodiversity and ecosystem services. Our work shares the purpose of Cai et al. (2014), but is otherwise complementary: while their work considers more finely partitioned land uses, 2 ours emphasizes the role of endogenous technological progress through R&D activities, and also allows population to respond to changes in agricultural productivity through endogenous fertility. As we consider responses to agricultural productivity shocks, our work also relates to an extensive microeconometric literature that studies variability in agricultural productivity. In this area, one line of research exploits exogenous variations in rainfall to quantity the impact of TFP variations on outcomes in the agricultural sector (see notably Jayachandran, 2006; Di Falco and Chavas, 2008). Close to our main topic of interest, Auffhammer et al. (2006) have shown that rainfall variability affects the choice of cropland area under cultivation at the farm level. 3 As Auffhammer and Schlenker (2014) note, one limitation of these reduced-form studies is that long-run effects and feedback mechanisms (e.g. general equilibrium) are difficult to identify from the data. From this perspective, our structural empirical model provides novel perspectives on these issues, accounting for a number of macro-level interrelationships between endogenous outcomes, and quantifying how these jointly respond to negative agricultural supply shocks. 2 More specifically, Cai et al. (2014) consider the allocation of land to commercially managed forests (with many different stock variables capturing different forest vintages) and to biofuel crops. Forest products and energy are consumed by households. Non-converted natural land generates ecosystem services, which are also valued by households. 3 See also Schlenker and Roberts (2009) and Fezzi and Bateman (2015) on the role of temperature variability. 7

10 It is also important to stress that our aggregate global representation has its limitations, and abstracts from a number of dimensions that have been discussed in the literature. First, by construction, our model cannot inform spatial aspects of development, which include international markets for agricultural commodities, and trade. In particular, because the world as a whole is modeled as one region, factors are mobile in our framework, and openness to trade is only implicit. Our model is, however, consistent with a multiregional model with trade in which the expansion of agricultural land is incentivized through changes in international commodity prices. For example, a negative agricultural supply shock in a given region may not have an impact on population or agricultural land area in that particular region, but if the shock is large enough to have macro-level repercussions (as we do assume in our work), it will cause an increase in world agricultural prices. This would in turn affect outcomes in price-sensitive regions (typically developing regions), including fertility choices and agricultural land expansion. 4 This is consistent with Burgess and Donaldson (2010) and Costinot et al. (2016) for example, who emphasize the role of interregional price signals in the allocation of resources, as well as the literature that uses detailed numerical trade models of agricultural production, mentioned above. Second, our model does not capture more complex institutional dimensions of growth and food production that have been discussed elsewhere in the literature. One example is related to political dynamics at work in the presence of agricultural output variability. Using data from Sub- Saharan Africa, Brückner and Ciccone (2011) suggest that negative agricultural supply shocks may provide a window of opportunity for improved democracy. In turn, improved democracy would be expected to have a positive impact on economic growth (Acemoglu et al., 2017). In our model, while negative shocks do lead to faster TFP growth, the channel through which TFP increases (labor-intensive R&D) is inconsistent with an institutional view of growth. Similarly, an extensive literature studies how local scarcities induce conflict and migration (see e.g. Prieur and Schumacher, 2016, for an overview); the associated welfare costs are only implicit in our highly aggregated representation of the world. Therefore, while our empirical framework brings 4 Note that our model accounts for the fact that remaining reserve lands are likely to be less productive, compared to land already under cultivation. We come back to this below. 8

11 together several well-established strands of economic research to provide novel insights into the impacts of negative agricultural productivity shocks, its limitations ought to be kept in mind. 3 The model This section first summarizes the key components of the model. Second, we present the simulationbased structural estimation procedure. Third, we explain how we introduce stochastic shocks to the evolution of agricultural productivity The economy Manufacturing production and agriculture A manufacturing sector produces the traditional consumption bundle in one-sector models, with aggregate output Y t,mn at time t given by: Y t,mn = A t,mn K ϑ t,mnl 1 ϑ t,mn, (1) where A t,mn is TFP in manufacturing, K t,mn is capital and L t,mn is the workforce. 6 The share of capital is set to 0.3, which is consistent with Gollin (2002), for example. Agricultural output Y t,ag is given by a flexible nested constant elasticity of substitution (CES) function (see Kawagoe et al., 1986; Ashraf et al., 2008), in which the lower nest is Cobb-Douglas in capital and labor, and the upper nest trades off the capital-labor composite with the land input X t : ( ) Y t,ag = A t,ag [(1 θ X ) K θ K t,ag L 1 θ σ 1 K σ t,ag ] σ + θ X X σ 1 σ 1 σ t, (2) 5 As noted above, Lanz et al. (forthcoming) provides a comprehensive motivation for the structure of the model, analytical results on the evolution of population and land, discussion of the selection and estimation of the parameters, as well as ensuing baseline projections from 2010 onwards. Extensive sensitivity analysis is also reported, showing that the baseline projections are robust to a number of changes to the structure of the model, which comes from the fact that we estimate the model over a relatively long horizon. The GAMS code for the model, replicating the baseline runs reported here, is available on Bruno Lanz s website. 6 Note that under the assumption that technology is Hicks-neutral, the Cobb-Douglas functional form is consistent with long-term empirical evidence reported in Antràs (2004). 9

12 where σ determines substitution possibilities between the capital-labor composite and land. Following empirical evidence reported in Wilde (2013), representing long-term substitution possibilities between land and other factors in agriculture, we set σ = 0.6. We further set the share parameters θ X = 0.25 and θ K = 0.3 based on data from Hertel et al. (2012). Innovations and technological progress The evolution of sectoral TFP is given by (in the absence of negative productivity shocks, discussed below): A t+1,j = A t,j (1 + ρ t,j S), j {mn, ag}, (3) where j is an index for sectors (here mn is manufacturing and ag is agriculture), S = 0.05 is the maximum aggregate growth rate of TFP each period (based on Fuglie, 2012), and ρ t,j [0, 1] measures the arrival rate of innovations, i.e. how much of the maximum growth rate is achieved each period. TFP growth in the model, which is driven by ρ t,j, is a function of labor allocated to sectoral R&D: ρ t,j = λ j ( Lt,Aj N t ) µj, j {mn, ag}, where L t,aj is labor employed in R&D for sector j, λ j is a productivity parameter (normalized to 1 to ensure that TFP growth is bounded between 0 and S) and µ j (0, 1) is an elasticity. The parameters µ mn and µ ag are structurally estimated and capture the extent of decreasing returns to labor in R&D (e.g. duplication of ideas among researchers; Jones and Williams, 2000). Expressions (3) and (4) represent a discrete-time version of the original model by Aghion and Howitt (1992), in which the arrival of innovations is modeled as a continuous-time Poisson process. 7 One key departure from Aghion and Howitt (1992), however, is that the growth rate of TFP is a function of the share of labor allocated to R&D. This representation, which is also 7 We implicitly make use of the law of large number to integrate out random arrival of innovation over discrete time intervals. 10

13 discussed in Jones (1995a) and Chu et al. (2013), is consistent with microfoundations of more recent product-line representations of technological progress (e.g. Dinopoulos and Thompson, 1998; Peretto, 1998; Young, 1998), in which individual workers are hired by R&D firms and entry of new firms is allowed (Dinopoulos and Thompson, 1999). One feature of such representations, and therefore of ours, is the absence of the population scale effect, in other words a positive equilibrium relationship between the size of the population and technological progress. 8 Indeed, over time the entry of new firms dilutes R&D inputs and neutralizes the scale effect, and in equilibrium aggregate TFP growth is proportional to the share of labor in R&D (see Laincz and Peretto, 2006). Population dynamics Population in the model represents the stock of effective labor units N t and evolves according to the standard motion equation: N t+1 = N t (1 + n t δ N ), N 0 given, (4) where 1/δ N captures the expected working lifetime, which is set to 45 years (hence δ N = 0.022), and increments to the labor force n t N t are a function of labor L t,n allocated to rearing and educating children: n t N t = χ t L t,n. In this setting, 1/χ t is a measure of the time (or opportunity) cost of effective labor units, and a significant component of this cost is education. As mentioned earlier, empirical evidence suggests a complementarity between human capital and technology (e.g. Goldin and Katz, 1998), and 8 Note that Boserup (1965) and Kremer (1993) use the population scale effect to explain the sharp increase of productivity growth following stagnation in the pre-industrial era, and it is also present in unified growth theory models by Galor and Weil (2000) and Jones (2001) among others. Empirical evidence from more recent history, however, is at odds with the scale effect (e.g. Jones, 1995b; Laincz and Peretto, 2006). The fact that it is absent from our model is important, because population is endogenous, so that accumulating population could be exploited to artificially increase long-run growth. 11

14 we specify the cost of children as an increasing function of the economy-wide level of technology: χ t = χl ζ 1 t,n /Aω t, where χ > 0 is a productivity parameter, ζ (0, 1) is an elasticity representing scarce factors required in child rearing, A t is an output-weighted average of sectoral TFP, and ω > 0 measures how the cost of children increases with the level of technology. The parameters determining the evolution of the cost of increments to the labor force (χ, ζ and ω) are estimated as described below. We show analytically in Lanz et al. (forthcoming) that this representation of the cost of children is consistent with the more comprehensive model of Galor and Weil (2000), in which education decisions are explicit and the relationship between technology and human capital arises endogenously. More specifically, in our model the accumulation of human capital is implicit, as it is functionally related to the contemporaneous level of technology. Like in Galor and Weil (2000), however, technological progress raises the cost of children by inducing higher educational requirements, and is therefore an important driver of the demographic transition. In other words, the positive relationship between technology and the cost of effective labor units implies that, over time, the quality of children (measured by their level of education) required to keep up with technology is favored over the quantity of children, leading to a decline of fertility and population growth. In addition to the opportunity cost of time, there is an additional cost to population increments through the requirement that sufficient food must be produced. Formally, we follow Strulik and Weisdorf (2008) and make agricultural output a necessary condition to sustain the contemporaneous level of population (see also Vollrath, 2011; Sharp et al., 2012, for similar approaches): Y ag t = N t f t, (5) where f t is per-capita demand for food. In order to include empirical evidence about the income 12

15 elasticity of food demand, we further specify f = ξ ( Yt,mn N t ) κ, with income elasticity of food demand κ = 0.25 reflecting estimates reported in Thomas and Strauss (1997) and Beatty and LaFrance (2005). We further calibrate the parameter ξ = 0.4 so that aggregate food demand in 1960 is about 15 per cent of world GDP (as per data reported in Echevarria, 1997). Agricultural land conversion Land is a necessary input to agriculture, and agricultural land X t has to be converted from a fixed stock of natural land reserves (X) by applying labor L t,x. 9 In our model, land is therefore treated as a scarce form of capital, and we write the motion equation for agricultural land as: X t+1 = X t (1 δ X ) + ψ L ε t,x, X 0 given, X t X, (6) where the parameters ψ > 0 and ε (0, 1) are structurally estimated. Through equation (6), we allow converted land to revert back to its natural state over a fifty-year time frame (i.e. δ X = 0.02). Note also that an important implication of (6) is that, as labor is subject to decreasing returns in land-conversion activities, the marginal cost of land conversion increases with X t. Intuitively, this captures the fact that the most productive plots are converted first, whereas additional land might be less amenable to exploit for agricultural production. An implication is that the cost associated with bringing marginal plots into production because of uncertainty is higher than the cost of converting land earlier in the development process. Households preferences and savings In the tradition of Barro and Becker (1989), household preferences are defined over own con- 9 Note that aside from the space needed to grow the food, the model does not quantify the demand for space by agents in the model, such as industrial use to produce manufactured goods, or residential use to accommodate the growing population. While this sort of land-use competition is certainly important at a local level, we abstract from that to focus on an aggregate global representation of development. 13

16 sumption of a (composite) manufactured good, denoted c t, the level of fertility n t and the utility that surviving members of the family will enjoy in the next period U i,t+1. Given survival probability 1 δ N, and simplifying assumptions that (i) children are identical and (ii) parents value their own utility in period t+1 the same as their children s (see Jones and Schoonbroodt, 2010), the utility function of a representative household is defined recursively as: U t = c1 γ t 1 + β[(1 δ N ) + n t ] 1 η U t+1, 1 γ where γ = 2 reflects an intertemporal elasticity of substitution of 0.5 (e.g. Guvenen, 2006), β = 0.99 is the discount factor and η is an elasticity determining how the utility of parents changes with the number of surviving members of the household. As we show in Lanz et al. (forthcoming), it is straightforward to express preferences from the perspective of the dynastic household head, yielding the following dynastic utility function: U 0 = t=0 β t N 1 η t c 1 γ t 1 1 γ, (7) and we set η = This implies that altruism towards surviving members of the dynasty remains almost constant as the number of survivors increases. It makes the household s objective close to the standard Classical Utilitarian welfare function. As in the multi-sector growth model of Ngai and Pissarides (2007), manufacturing output can either be consumed or invested into a stock of physical capital: Y t,mn = N t c t + I t, (8) where N t c t and I t measure aggregate consumption and investment respectively. The motion equation for capital is given by: K t+1 = K t (1 δ K ) + I t, K 0 given, (9) where δ K = 0.1 is the yearly rate of capital depreciation (Schündeln, 2013). 14

17 3.2 Structural estimation of the model A schematic representation of the model is provided in Figure 2. We formulate the model as a social-planner problem, selecting paths for investment I t, and allocating labor L t,j and capital K t,j across activities in order to maximize intertemporal welfare (7) subject to technological constraints (1), (2), (3), (4), (5) (6), (8), (9) and feasibility conditions for capital and labor: K t = K t,mn + K t,ag, N t = L t,mn + L t,ag + L t,amn + L t,aag + L t,n + L t,x. The constrained non-linear optimization problem associated with the planner s program is solved numerically by searching for a local optimum of the objective function (the discounted sum of utility) subject to the requirement of maintaining feasibility as defined by the constraints of the problem. 10 We apply simulation methods to structurally estimate parameters determining the cost of fertility (χ, ζ, ω), labor productivity in R&D (µ mn,ag ) and labor productivity in land conversion (ψ, ε). In practice, we first calibrate the initial value of the state variables to match 1960 data, so that the model is initialized in the first year of the estimation period. For each parameter to be estimated from the data, we define bounds for possible values (0.1 and 0.9 for elasticities and 0.03 and 0.3 for labor productivity parameters) and simulate the model for a randomly drawn set of 10,000 vectors of parameters. We then formulate a minimum distance criterion, which compares observed time series for world GDP (Maddison, 1995; Bolt and van Zanden, 2013), population (United Nations, 1999b, 2013), cropland area (Goldewijk, 2001; Alexandratos and Bruinsma, 2012) and sectoral TFP (Martin and Mitra, 2001; Fuglie, 2012) with trajectories simulated from the model. 11 In the model these data correspond to Y t,mn +Y t,ag, 10 The numerical problem is formulated in GAMS and solved with KNITRO (Byrd et al., 1999, 2006), a specialized software programme for constrained non-linear programs. Note that this solution method can only approximate the solution to the infinite horizon problem, as finite computer memory cannot accommodate an objective with an infinite number of terms and an infinite number of constraints. However, for β < 1 only a finite number of terms matter for the solution, and we truncate the problem to the first T = 200 periods without quantitatively relevant effects for our results. 11 Note that TFP growth estimates are subject to significant uncertainty, and we conservatively assume that it declines from 1.5 per cent between 1960 and 1980 to 1.2 per cent between 1980 and 2000, and then stays at 1 per cent over the last decade. 15

18 Figure 2: Schematic representation of the model Welfare Manufacturing technology A MN Manufacturing output Y MN Savings decision C / I Population N Labor allocation L MN, L AG, L A,MN, L A,AG, L X, L N HUMAN CAPITAL CONSTRAINT Capital allocation K MN, K AG Capital stock K Agricultural technology A AG Agricultural land area X Agricultural output Y AG LEGEND: Stock variables Control variables FOOD REQUIREMENTS CONSTRAINT 16

19 N t, X t, A t,mn and A t,ag respectively. Thus, formally, for each vector of parameters and associated model solution, we compute: [ (Zk,τ Z k,τ ) 2 / τ τ k Z k,τ ], (10) where Z k,τ denotes the observed quantity k at time τ and Z k,τ is the corresponding value simulated by the model. By gradually refining the bounds of each parameter, we converge to a vector of parameters that minimizes objective (10). We find that the model closely fits the targeted data; the resulting vector of estimates and fitted trajectories over the estimation period are reported and briefly discussed in the Appendix (see also Lanz et al., forthcoming, for an extensive discussion of the estimation results). At this stage it is important to note that the social planner representation is mainly used as a tool to make structural estimation of the model tractable: we rationalize the data as if it had been generated by a social planner. Thus market imperfections prevailing over the estimation period will be reflected in the parameters that we estimate from observed trajectories, and will thus be reflected in the baseline simulations of the model (i.e. using the model to extrapolate the behavior of the system observed over the past fifty years). 12 But given the estimated technological parameters, simulations with the model away from the baseline will reflect a socially optimal allocation of resources. 3.3 Introducing stochastic shocks to agricultural productivity In the basic formulation of the model, which is used for estimating the parameters over the period , the evolution of sectoral TFP is deterministic and depends on the share of labor employed in sectoral R&D activities. We now study the evolution of the system beyond 12 Because there are externalities in the model, most notably in R&D activities (see Romer, 1994, for example) the optimum determined by the social planner solution will differ from a decentralized allocation. Thus if we were able to estimate the parameters using a decentralized solution method, a different set of estimates would be required to match observed trajectories over the estimation period. As shown by Tournemaine and Luangaram (2012) in the context of similar model (without land), however, quantitative differences between centralized and decentralized solutions are likely to be small. 17

20 2010, and introduce stochasticity in how agricultural TFP evolves over time. Specifically, it is assumed that technological progress in agriculture is subject to stochastic shocks of size ɛ > 0 that occur with probability p. Conversely with probability 1 p there is no shock to agricultural productivity (hence ɛ = 0) and the evolution of TFP occurs as per the deterministic specification described above. Both p and ɛ are assumed to be known by the planner, thus the situation is one of pure risk. 13 Formally, equation (3) describing the evolution of agricultural productivity is augmented with a non-negative term, which represents the possibility that agricultural TFP may not follow the functional trajectory we have postulated: Ã t+1,ag,s = Ãt,ag,s (1 + ρ t,ag,s S ɛ t+1,s ), (11) where ɛ t+1,s captures the specific realization of the shock in state of the world s, and we index all variables by s to capture the fact that they are conditional on a specific sequence of ɛ t,s over time. A stochastic shock affects outcomes in period t + 1, while the planner only observes the outcome after allocating resources in period t. We further assume that the planner is an expected utility maximizer, weighting welfare in the different states of the world by its respective probability. The ensuing objective function is then: W = s p s t=0 β t N 1 η t,s c 1 γ t,s 1 1 γ, (12) with s p s = Even though this stochastic structure is quite simple, the number of possible states of the world in each period grows at 2 t. In turn, because the model is formulated as a non-linear 13 We note that the probability of negative shocks and their size might be a function of agricultural activities. In a companion paper (Lanz et al., 2016), we discuss how the scale of modern agriculture may affect such negative feedback effect, focusing on the expected impact of negative shocks over time rather than on stochastic occurrences. In the present paper, however, we focus on a more general exogenous source of uncertainty, in which the probability and size of shocks is fixed. 14 Note that this formulation implies the standard assumption that markets are complete, both over time and across states of the world. 18

21 optimization problem, this implies that the number of variables that needs to be computed over the whole horizon increases exponentially. 15 Given that the dimensionality of the decision problem grows with the set of possible states of the world, we make two further simplifications. First, we solve the model from 2010 onwards using two-year time steps (instead of yearly time steps). This significantly reduces the number of variables that needs to be computed, without significantly affecting the resulting trajectories. 16 Second, we consider shocks in only three time periods, which is sufficient to illustrate the mechanisms at work. The shock we consider is a 10 per cent probability that agricultural production declines by 5 per cent each year over two years. This is in the range implied by Figure 1, and is also broadly consistent with changes in productivity discussed in Nelson et al. (2014b) and Cai et al. (2014). Hence, starting the simulation in 2010, we assume that the first realization of the shock may occur after 2016 allocation decisions have been made, so that effects are felt in In the bad state of the world, which occurs with a probability of 10%, agricultural TFP is (1 0.05) 2 = 0.9 of that prevailing in the good state of the world. In expected value terms, the shock is thus roughly equivalent to a one per cent decrease in TFP over two years. The same shock can then occur in 2018, with effects felt in 2020, and in 2020, with effects felt in To summarize, we initialize the model in 2010, and negative TFP shocks can occur in 2016, 2018 and 2020, with effects being felt in subsequent periods. After 2022, no more shocks occur and the problem becomes deterministic (conditional on the state of the world in which the planner happens to be). Of course, the results would remain qualitatively similar if we were to consider the reoccurence of shocks beyond 2020, so that it is relatively easy to see how our results would generalize. 15 More specifically, as the planner faces a dynamic problem, optimal decisions in each time period are conditional on the history of shocks (i.e. where he is in the exponentially-growing uncertainty tree), and the planner maximizes the expected utility of his decisions over the remaining event tree. Thus states of the world sharing a common parent node will share decision variables until the subsequent realization of the productivity shock, and diverge thereafter, so that computational requirements increase. 16 Increasing the time-steps to evaluate the choice of the controls implies some small differences in optimal paths relative to the solution using one-year time steps. Another approach would be to formulate the problem recursively and solve it with dynamic programming methods. This approach is, however, subject to dimensionality restrictions in terms of the number of state variables that can be included. 19

22 4 Results: Optimal control and simulations This section provides the main results from solving the stochastic control problem. First, we describe the particular agricultural productivity scenarios that we focus on. Second, we report implied trajectories for agricultural technology, agricultural land, population and welfare. 4.1 Scenario description To evaluate the socially optimal response to agricultural productivity risk, we contrast trajectories resulting from four different situations. First, we consider a case in which no shocks to agricultural TFP will occur, and the planner knows this for sure. This represents our baseline, as reported in Lanz et al. (forthcoming). Values for selected variables are reported in Table 1. World population starts at just below 7 billion in 2010 and grows to 8.5 billion by 2030, a 20 per cent increase. At the same time, cropland area increases by 70 million hectares, or 5 per cent. These figures are broadly consistent with the latest population projections of the United Nations (2015) and with land-use projections by FAO, reported in Alexandratos and Bruinsma (2012), and AgMIP, reported in Schmitz et al. (2014). The growth rate of agricultural TFP starts at 0.9 per cent per year in 2010 and declines over time, which is rather conservative compared with the assumptions used in Alexandratos and Bruinsma (2012). Importantly, these figures represent projections from the fitted model and are thus informed by the evolution of agricultural TFP from 1960 to 2010, as the estimated model essentially projects forward the pace of development that has been observed in recent history. The second situation we consider is also deterministic. We assume that shocks occur in 2016, 2018 and We label this scenario In the period just following each of the three shocks, agricultural TFP is exogenously brought down by 10 per cent, although the planner anticipates each shock and can reallocate resources relative to the baseline. In the third scenario, labeled expected value, the planner allocates resources taking into account the expected value of the TFP reduction. In other words, he takes into account the risk of a 10 per cent reduction in TFP each decision period, but weights that reduction by the associated probability of 10 per cent. Thus, agricultural TFP growth in each decision period is 20

23 Table 1: Deterministic no shocks scenario: Baseline values for selected variables World population (billion) Cropland area (billion hectares) Yearly agricultural TFP growth rate Per-capita consumption (thousand intl. dollars) exogenously brought down by around one percentage point. This scenario amounts to analyzing the allocation decisions of a risk-neutral planner, and where the realization of the shock happens to be exactly the expected value of the shock. Finally, we compute trajectories that maximize expected utility. In this situation, the planner is risk-averse (relative risk aversion is set to γ = 2). He takes into account the risk that agricultural TFP may decline, and what this entails for social welfare. A key point is that allocation decisions are contingent on the realized state of the world. In other words, after each decision period in which the risk is realized, the decision tree branches out, and the planner makes allocation decisions contingent on being in a particular node in the uncertainty tree. By construction, there are then 2 3 = 8 possible states of the world in 2030, and thus the same number of stochastic scenarios for an expected-utility maximizing planner (we label each stochastic scenario according to the years in which TFP shocks are realized). 4.2 Agricultural technology paths Figure 3 shows the paths for agricultural TFP under alternative scenarios. Starting with the deterministic scenarios, which are displayed in panel (a), agricultural TFP grows linearly at around one per cent per year (and falling slightly) under the best-case no shocks scenario. Under the deterministic expected value path, TFP grows at a lower pace from 2016 to 2020, reflecting the expected value of the negative shocks. But before 2016 TFP grows ever so slightly quicker in the expected value scenario, because the planner knows that small negative shocks will occur from 2016 to 2020 and makes provisions for them (see below). This anticipatory 21